## CRISTOFORO CAFFI

PhD Graduate**PhD program:**: XXXIII

**supervisor**: A. De Sole

**advisor**: P. Papi

**Thesis title:**Conformal Embeddings in Basic Lie Superalgebras

In this thesis we deal with some aspect concerning embeddings of regular subalgebras $\g^0\subset\g$, where $\g$ is a basic Lie superalgebra.
A first problem is to find a criterion to determine whether a regular subalgebra is maximal or not, similarly to what has been done for semisimple Lie algebras. We discovered that the the maximal equal rank subalgebras are obtained by taking all possible affine diagrams and deleting a dot corresponding to a simple root with prime coefficient.
Then, for every equal rank regular subalgebra $\g^0\subset\g$, we use the AP criterion to find the conformal levels of the embeddings, that is, the values of $k$ such that the Virasoro vector of the vertex subalgebra $\mathcal{V}_k(\g)$ of $V_k(\g)$ generated by $\g^0$ is equal to the Virasoro vector of $V_k(\g)$.
We also describe some cases in which, using the fusion rule argument, we can decompose the simple vertex algebra $V_k(\g)$ as $\mathcal{V}_k(\g)$-module.